Entropy and Global Existence for Hyperbolic Balance Laws

This paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables. We propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state. In addition, we show that a system of balance laws satisfies a Kawashima condition if and only if its first-order approximation, namely the hyperbolic-parabolic system derived through the Chapman-Enskog expansion, satisfies the corresponding Kawashima condition. The result is then applied to Bouchuts discrete velocity BGK models approximating hyperbolic systems of conservation laws.     

Dissipative Structure and Entropy for Hyperbolic Systems of Balance Laws

In this paper, we introduce an entropy condition for hyperbolic systems of balance laws. Under this condition, we use the Chapman-Enskog expansion to derive the corresponding viscous conservation laws. Further structural conditions are discussed in order to develop (local and global) existence theories for the balance laws and viscous conservation laws.Acknowledgement The research of S. KAWASHIMA was partially supported by Grant-in-Aid for Scientific Research (No. 14340047), The Ministry of Education, Culture, Science and Technology, Japan. The reseach of W.-A. YONG was supported by the Deutsche Forschungsgemeinschaft through the Schwerpunktprogramm ANumE and SFB 359 at the University of Heidelberg and by the European TMR-Network Hyperbolic and Kinetic Equations.     

Coleman—Noll关系式与熵函数存在条件

本文以Coleman-Noll关系式(特指使Clausius-Duhem不等式成立的充要条件)为基础,对内变量理论、粘弹性及塑性流动理论等情形研究了熵函数存在条件的具体形式以及熵函数应具有的表达式等相关问题.     

Ricci Curvature of Finite Markov Chains via Convexity of the Entropy

We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry–Émery and Otto–Villani. Further, we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.     

Existence of Global Entropy Solutions of a Nonstrictly Hyperbolic System

In this paper we use the theory of compensated compactness coupled with some basic ideas of the kinetic formulation by Lions, Perthame, Souganidis & Tadmor [LPS, LPT] to establish an existence theorem for global entropy solutions of the nonstrictly hyperbolic system (1).     

Global BV Entropy Solutions and Uniqueness for Hyperbolic Systems of Balance Laws

We consider the Cauchy problem for n×n strictly hyperbolic systems of nonresonant balance laws each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that and are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation as limits of special wave-front tracking approximations for which the source term is localized by means of Dirac masses. Moreover, we give a characterization of the resulting semigroup trajectories in terms of integral estimates.     

Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy

We consider the Cauchy problem for a general one-dimensional n×n hyperbolic symmetrizable system of balance laws. It is well known that, in many physical examples, for instance for the isentropic Euler system with damping, the dissipation due to the source term may prevent the shock formation, at least for smooth and small initial data. Our main goal is to find a set of general and realistic sufficient conditions to guarantee the global existence of smooth solutions, and possibly to investigate their asymptotic behavior. For systems which are entropy dissipative, a quite natural generalization of the Kawashima condition for hyperbolic-parabolic systems can be given. In this paper, we first propose a general framework for this kind of problem, by using the so-called entropy variables. Then we go on to prove some general statements about the global existence of smooth solutions, under different sets of conditions. In particular, the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension. Our main tools will be some refined energy estimates and the use of a suitable version of the Kawashima condition.     

Convexity and symmetrization in relativistic theories

There is a strong motivation for the desire to have symmetric hyperbolic field equations in thermodynamics, because they guarantee well-posedness of Cauchy problems. A generic quasi-linear first order system of balance laws — in the non-relativistic case — can be shown to be symmetric hyperbolic, if the entropy density is concave with respect to the variables. In relativistic thermodynamics this is not so.This paper shows that there exists a scalar quantity in relativistic thermodynamics whose concavity guarantees a symmetric hyperbolic system. But that quantity — we call it — — is not the entropy, although it is closely related to it. It is formed by contracting the entropy flux vector — h^a with a privileged time-like congruence .It is also shown that the convexity of h plus the requirement that all speeds be smaller than the speed of light c provide symmetric hyperbolic field equations for all choices of the direction of time.At this level of generality the physical meaning of —h is unknown. However, in many circumstances it is equal to the entropy. This is so, of course, in the non-relativistic limit but also in the non-dissipative relativistic fluid and even in relativistic extended thermodynamics for a non-degenerate gas.     

结构信息熵与极大熵原理

文中首先从数学上证明了由结构应变能描述的信息熵是定义在凸集上的凸函数；又以静定杆系结构为对象，论证并讨论了极大熵原理。最后通过对静定与静不定杆系结构的数值模拟，揭示了结构的熵函数与杆系截面积之间的关系。     

Displacement Convexity and Minimal Fronts at Phase Boundaries

We show that certain free energy functionals that are not convex with respect to the usual convex structure on their domain of definition are strictly convex in the sense of displacement convexity under a natural change of variables.We use this to show that, in certain cases, the only critical points of these functionals are minimizers. This approach based on displacement convexity permits us to treat multicomponent systems as well as single component systems. The developments produce new examples of displacement convex functionals and, in the multi-component setting, jointly displacement convex functionals.     

Convexity Criteria and Uniqueness of Absolutely Minimizing Functions

We show that an absolutely minimizing function with respect to a convex Hamiltonian \({H : \mathbb{R}^{n} \rightarrow \mathbb{R}}\) is uniquely determined by its boundary values under minimal assumptions on H. Along the way, we extend the known equivalences between comparison with cones, convexity criteria, and absolutely minimizing properties, to this generality. These results perfect a long development in the uniqueness/existence theory of the archetypal problem of the calculus of variations in L^∞.     

Canonical and Anti-Canonical Transformations Preserving Convexity of Potentials

The aim of the paper is to characterize transformations that preserve the potential structure of a relationship between dual variables. The first step consists in deriving a geometric definition of the condition for the existence of a potential. Having at hand this formulation, it becomes clear that the canonical similitudes represents the class of transformations that preserves the potential form of a relationship. Next, we derive the conditions under which canonical similitudes preserve the convexity of the potential or change it into concavity. This new class of transformations can be viewed as a generalization of the Legendre-Fenchel transformation. These concepts are applied to the Hooke constitutive relationship.     

Hyperbolic evolution groups and dichotomic evolution families

Recently, Ben-Artzi and Gohberg [2] used the concept ofC ₀-semigroups in order to characterize the existence of dichotomies for nonautonomous differential equations on _n. A similar task was performed by Latushkin and Stepin [11] for dichotomies of linear skew-product flows. In this paper we will useC _o-semigroups to characterize existence of dichotomies for strongly continuous evolution families (U(t,s)) _t.s on Hilbert and Banach spaces. Under an exponential growth condition we show that the concepts of hyperbolic evolution groups and exponentially dichotomic evolution families are equivalent.     

Invariants of Hyperbolic Equations: Solution of the Laplace Problem

This paper gives a solution of the Laplace problem, which consists of finding all invariants of the hyperbolic equations and constructing a basis of the invariants. Three new invariants of the first and second orders are found, and invariantdifferentiation operators are constructed. It is shown that the new invariants, together with the two invariants detected by Ovsyannikov, form a basis such that any invariant of any order is a function of the basis invariants and their invariant derivatives.     

Entropy Dissipation and Long-Range Interactions

We study Boltzmann's collision operator for long-range interactions, i.e., without Grad's angular cut-off assumption. We establish a functional inequality showing that the entropy dissipation controls smoothness of the distribution function, in a precise sense. Our estimate is optimal, and gives a unified treatment of both the linear and the nonlinear cases. We also give simple and self-contained proofs of several useful results that were scattered in previous works. As an application, we obtain several helpful estimates for the Cauchy problem, and for the Landau approximation in plasma physics. Accepted: November 23, 1999     

Nonlinear Hyperbolic Systems: Nondegenerate Flux,Inner Speed Variation,and Graph Solutions

We study the Cauchy problem for general nonlinear strictly hyperbolic systems of partial differential equations in one space variable. First, we re-visit the construction of the solution to the Riemann problem and introduce the notion of a nondegenerate (ND) system. This is the optimal condition guaranteeing, as we show it, that the Riemann problem can be solved with finitely many waves only; we establish that the ND condition is generic in the sense of Baire (for the Whitney topology), so that any system can be approached by a ND system. Second, we introduce the concept of inner speed variation and we derive new interaction estimates on wave speeds. Third, we design a wave front tracking scheme and establish its strong convergence to the entropy solution of the Cauchy problem; this provides a new existence proof as well as an approximation algorithm. As an application, we investigate the time regularity of the graph solutions (X,U) introduced by LeFloch, and propose a geometric version of our scheme; in turn, the spatial component X of a graph solution can be chosen to be continuous in both time and space, while its component U is continuous in space and has bounded variation in time.